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| Mirrors > Home > MPE Home > Th. List > Mathboxes > naryrcl | Structured version Visualization version GIF version | ||
| Description: Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.) | 
| Ref | Expression | 
|---|---|
| naryfval.i | ⊢ 𝐼 = (0..^𝑁) | 
| Ref | Expression | 
|---|---|
| naryrcl | ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-naryf 48553 | . 2 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥)))) | |
| 2 | 1 | elmpocl 7675 | 1 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 (class class class)co 7432 ↑m cmap 8867 0cc0 11156 ℕ0cn0 12528 ..^cfzo 13695 -aryF cnaryf 48552 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-dm 5694 df-iota 6513 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-naryf 48553 | 
| This theorem is referenced by: naryfvalelfv 48558 0aryfvalelfv 48561 fv1arycl 48563 1arymaptfv 48566 fv2arycl 48574 2arymaptfv 48577 | 
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